= Harmonic quadrant solution from tangential boundary derivatives
For a <harmonic function> in the quadrant with boundary derivatives $u_y(0,y)=g_1(y)$ and $u_x(x,0)=g_2(x)$, integrating the data to equal corner values and applying the <Schwarz integral formula> after the <conformal map> $w=z^2$ gives
$$
u_z(z)=\frac z{\pi i}\left[\int_0^\infty\frac{g_2(r)}{r^2-z^2}\,dr+\int_0^\infty\frac{g_1(r)}{r^2+z^2}\,dr\right].
$$
These are <tangential boundary derivatives>, so uniqueness requires an additional normalization or growth restriction. A homogeneous contribution has the form $ih(z^2)/z$ with $h$ <holomorphic> in the <complex upper half-plane> and real on its boundary away from zero. For example, $u=xy$ has zero prescribed derivatives on both edges.
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